Finite Group Databasebeta

← #49120565

Subgroups — #49120565

Order 36224.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827207C211yesmaximal, characteristic
2#31826827C212yesmaximal, complemented
3#31826205C211yesmaximal, characteristic, complemented
4#31826204C212yesmaximal, complemented
5#497×C283C211yesmaximal, characteristic, complemented
6#22583599—21no—
7#22583598C2212yes—
8#22583596—22no—
9#22583580—22no—
10#22583578—21no—
11#22583533—22no—
12#22583530—22no—
13#22583521C2212yescomplemented
14C2×Q16×C283—21no—
15C2×SD16×C283—21no—
16C4⋊Q8×C283C2211yescharacteristic, complemented
17#124×C283C2211yescharacteristic
18#97×C283C2211yescharacteristic, complemented
19C23×D283—21no—
20C2×Dic566D412yes—
21C2×D1132—21no—
22C2×D1132D412yes—
23C2×C22⋊D283—22no—
24C2×C283⋊C8—22no—
25C22⋊Dic283—21no—
26(C2×D283)⋊C4—22no—
27(C2×D283)⋊C4—22no—
28(C2×D283)⋊C4—22no—
29C4×Dic283—21no—
30C2×Q8×C283—21no—
31C2×Q8×C283D411yescharacteristic
32C2×D4×C283D411yescharacteristic
33Q16×C283—42no—
34SD16×C283—42no—
35C2×C8×C283—21no—
36C4⋊C4×C283—22no—
37C22⋊C4×C283—22no—
38C42×C283C2311yescharacteristic
39C22×D283—22no—
40C22×D283—81no—
41C22×D283—82no—
42Dic566—42no—
43D1132—42no—
44D1132—81no—
45C22⋊D283—82no—
46C2×Dic283—22no—
47C283⋊C8—42no—
48C23×C283—21no—
49Q8×C283—41no—
50Q8×C283—41no—
51Q8×C283—41no—
52D4×C283—41no—
53C2×C4×C283—21no—
54C2×C4×C283—41no—
55C2×C4×C283C2×D412yes—
56C2×C4×C283C2×D411yescharacteristic
57C8×C283—22no—
58C22×C283—41no—
59C22×C283—42no—
60C22×C283#12011yescharacteristic
61C2×D283—42no—
62C2×D283—42no—
63C2×D283—42no—
64C2×D283—84no—
65C4×C283—21no—
66C4×C283—22no—
67C4×C283—41no—
68C4×C283—81no—
69Dic283—42no—
70C2×C283—81no—
71C2×C283#45311yescharacteristic
72C2×C283#45512yes—
73D283—82no—
74C283#165911yescharacteristic, complemented, Sylow
75#1659—2831nomaximal, Sylow
76H(3,𝔽4)—2831no—
77#497D28311yescharacteristic, complemented
78#489—2832no—
79#329—2831no—
80#329—2832no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D28311yescharacteristic, complemented
84#124—2832no—
85#124—5662no—
86#124—5661no—
87#124C2×D28311yescharacteristic
88#120—5662no—
89#120—5661no—
90#103—5662no—
91#102—5662no—
92#97—2832no—
93#97C2×D28311yescharacteristic, complemented
94C24—5661no—
95C2×Q8—21no—
96C2×Q8—2832no—
97C2×Q8C22⋊D28311yescharacteristic
98C2×D4—2832no—
99C2×D4—5661no—
100C2×D4—5662no—
101C2×D4C22⋊D28311yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—5662no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—5662no—
109C22⋊C4—5661no—
110C22⋊C4—5662no—
111C22⋊C4—5662no—
112C42—5661no—
113C42C22×D28311yescharacteristic
114C23—21no—
115C23—5662no—
116C23—22641no—
117C23—22642no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—11322no—
122D4—41no—
123D4—11322no—
124D4—22641no—
125D4—22642no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—5662no—
129C2×C4D4×D28312yes—
130C2×C4C2×C22⋊D28311yescharacteristic
131C8—22no—
132C8—11322no—
133C22—41no—
134C22—42no—
135C22—11322no—
136C22—11322no—
137C22—11322no—
138C22—22644no—
139C22#2258357811yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—11322no—
145C2—81no—
146C2—22642no—
147C2#3182663211yescharacteristic
148C2#3182664112yes—
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#497×C283C2
socleC22×C283#120
prime radical#31827207C2
hypercenter#497D283
hypocenterC283#1659
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #31826827C2semidirect
3. #31826205C2semidirect
4. #31826204C2semidirect
5. #497×C283C2semidirect
13. #22583521C22semidirect
16. C4⋊Q8×C283C22semidirect
18. #97×C283C22semidirect
74. C283#1659semidirect
77. #497D283semidirect
83. C4⋊Q8C2×D283semidirect
93. #97C2×D283semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#318272072
#318268272
#318262052
#318262042
#497×C2832
#1659283